can we interpret a sentence involving more than one 'iff' in the following way?

Question

If there are a sequence of 'iff' in a sentence, can we make a conclusion from the sentence by dropping all the 'iff' that lie after the first 'iff' and drop all the statements between the first statement and the last statement? For example, let $U=[u_{ij}]$ and $A=[a_{ij}]$ be $m$ x $n$ matrices. Can we make from the following sentence

$A+U=A$ iff $a_{ij}+u_{ij}=a_{ij}$ for all $1\leq{i}\leq{m}$ and $1\leq{j}\leq{n}$, which holds iff $u_{ij}=0$ for all $1\leq{i}\leq{m}$ and $1\leq{j}\leq{n}$

the conclusion $A+U=A$ iff $u_{ij}=0$ for all $1\leq{i}\leq{m}$ and $1\leq{j}\leq{n}$ ?

Answer 1

That's perfectly fine. Given statements $\Phi_j, j=1..k$ with $$\Phi_1 \Leftrightarrow \Phi_2 \Leftrightarrow \ldots \Leftrightarrow \Phi_k$$ You can write any $$\Phi_i \Leftrightarrow \Phi_j$$ with any choice of $i,j$ and chain as much as you like. This is because $\Leftrightarrow$ is an equivalence relation.